This unifying survey and synthesis responds to the considerable challenge related to the abundance of VRP variants and to the relatively few general classifications and analyses of these problems and solution methods. The survey underlines that, while few general and efficient metaheuristics were proposed in the literature for this important class of problems, MAVRPs naturally share many common features, and most heuristic strategies developed for specific problems can be applied to a broader range of VRP variants. Hence, we conducted this analysis from a general perspective detached from the particular characteristics of the VRP attributes, and adopted a synthetic approach providing the means to cope with the abundance of contributions. We analysed in detail sixty-four successful metaheuristics for fifteen well-studied MAVRPs, identifying the main concepts and algorithmic-design principles, and highlighting the winning strategies of many efficient metaheuristics for a wide variety of variants.
When considering state-of-the-art methods, we observed recurrent notions such as mix, variability, hybridisation, cooperation, diversity, multiplicity, as well as balance, equilibrium, trade-off. It appears that most successful metaheuristics are not determined
by a single factor but are the result of a good balance between several elements of method-ology: the use of different search spaces, variable neighbourhoods, mixed continuous and discontinuous search, short-, medium- and long-term memories, trade-off between diver-sification and intendiver-sification, cooperation and collective intelligence, hybridisation, and so on. In brief,in unity and diversity lies strength. The performance of those methods in-dicates that each element plays an important role. On the one hand, long-term memories, jumps, recombinations and, generally, advanced guidance mechanisms providing diversifi-cation and, when relevant, population-diversity management methods have the potential to make the search progress in the general “big rugged valley” of MAVRPs. On the other hand, short and medium-term memories and well-designed solution-improvement meth-ods provide the aggressive search capabilities to complete the refinement of solutions.
We also observed that a clever implementation of algorithms is a necessary condition to yield competitive and scalable methods. Neighbourhood pruning procedures (granu-larity, sequential searches) or memories on already evaluated routes, route segments, and moves, are necessary in many cases. Furthermore, one may notice that many algorithms rely on randomization and dedicate most of their computing time to evaluating various potential choices, without taking much advantage of history and already performed com-putations that may in many cases be profitably used. More intelligent guidance schemes have thus the potential to lead to performance improvements.
The research avenues for developing efficient MAVRP heuristics are numerous. We conclude the paper by summing up some open research questions. In the previous sec-tions, we identified a number of search-space, neighbourhoods, and trajectory choices leading to successful MAVRP metaheuristics. One may then ask to what extent these choices should depend upon the variant of the problem, and whether it is possible to identify desirable search spaces and neighbourhoods for some broad MAVRP classes. Of a similar nature are studies related to the definition of population-diversity metrics (e.g., what type of distance for MAVRPs) and management methods, and whether it should it be dependent upon the particular problem setting. Designing adequate and general neighbourhood pruning procedures for MAVRPs is another important issue of similar nature, which may also be stated in terms of making current mechanisms, e.g., granular and sequential search, efficiently applicable to a large variety of attributes and problem settings. Such algorithmic developments and proof-of-concept studies make up a very challenging research area.
The integration of diversification and the appropriate balance between intensification and diversification are critical factors for efficient MAVRP metaheuristics. This area is closely related to the development of advanced mechanisms to extract knowledge out of the explored search-space areas and to globally guide the metaheuristics. Links to the fields of hyper-heuristics and landscape analysis should also be more thoroughly explored.
As this survey illustrates, a number of metaheuristic families, tabu search, adaptive large neighbourhood search, and hybrid genetic algorithms, in particular, are widely acknowledged for their performance on a variety of MAVRPs. Given how differently these metaheuristics define and explore the search space, they are very likely to lead to extremely effective hybrid algorithms and parallel cooperative methods. This is an extremely rich and promising research field, particularly given the trend toward problem settings including a continuously increasing number of attributes and solution methods
capable of addressing these attributes simultaneously.
To conclude, more general-purpose solvers, capable of handling a wide range of MAVRPs, are necessary to efficiently address practical routing applications in a timely manner. Many research questions have been answered by personalizing algorithms for each particular variant and by case-by-case improvements. However, solving generically (e.g., using a single solver and parameter set) a wide range of MAVRPs requires a better understanding of the problem foundations and the methods. This unifying survey and synthesis is a step toward reaching these goals.
Acknowledgments
While working on this project, T.G. Crainic was the NSERC Industrial Research Chair in Logistics Management, ESG UQAM, and Adjunct Professor with the Department of Computer Science and Operations Research, Universit´e de Montr´eal, and the Depart-ment of Economics and Business Administration, Molde University College, Norway, M.
Gendreau was the NSERC/Hydro-Qu´ebec Industrial Research Chair on the Stochastic Optimization of Electricity Generation, MAGI, ´Ecole Polytechnique, and Adjunct Pro-fessor with the Department of Computer Science and Operations Research, Universit´e de Montr´eal.
Partial funding for this project has been provided by the Champagne-Ardenne Re-gional Council, France, the Natural Sciences and Engineering Council of Canada (NSERC), through its Industrial Research Chair and Discovery Grant programs, by our partners CN, Rona, Alimentation Couche-Tard, la F´ed´eration des producteurs de lait du Qu´ebec, the Ministry of Transportation of Qu´ebec, and by the Fonds qu´eb´ecois de la recherche sur la nature et les technologies (FQRNT) through its Team Research Project program.
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